Supplement to Lecture 12 (Oct. 24) & Lecture 13 (Oct. 29)
An exam question on competitive equilibrium may look something like this. We started solving this
question in class, on Oct. 24. Parts (d) and (e) will be covered on Oct. 29.
Consider a representative consumer who has preferences over consumption of a
(C) and leisure time (l) given by the following utility function: U (C;l) = lnC + 0:02 ▯ l:
Note that utility is concave in consumption and linear in leisure. The coe¢ cient on leisure in the
utility function indicates how much the consumer likes leisure. The consumer splits available time
h = 100 hours into work time, N , and leisure time, l: In addition to labor income, the consumer
also receives dividend income, ▯ > 0, and pays lump-sum taxes T > 0 to the government.
rm produces a
nal good which is sold to the consumer at the price of 1:
rm hires labor, N , and produces the
nal good using the following technology of production
▯▯ d▯ 1▯▯
Y = zK N where z is the total factor productivity, K is the capital stock which is
and ▯ is the capital income share. Consistent with Canadian data, we consider ▯ = 0:3: We also
let the value of the capital stock, K, equal approximately 38:94, so that K ▯ = 38:94 0:3= 3: The
production function then becomes
Y = 3 ▯ z ▯ N
Note that we will not consider a speci
c value for the productivity parameter z. We will work
with symbol z instead, so that we can analyze model predictions to a change in TFP, z:
The government uses the tax revenues to
nance its expenditures G. We generally think of
G as an exogenous variable. For any level of G, it is possible to write government spending as
a fraction of total output produced: G = g ▯ Y; where g is a fraction (say 10%). In solving our
model, we will work with G = g ▯ Y because this will keep our algebraic derivations simple. Note:
If government spending is fraction g of total output, then consumption is fraction 1 ▯ g of total
output. Recall expenditure identity: C + G = Y ) C + gY = Y ) C = Y (1 ▯ g):
(a): Give a de
nition of an endogenous variable and an exogenous variable.
ne a competitive equilibrium in this example. Be speci
c about which variables are
endogenous and exogenous; who solves what problem; write down everyone s problem explicitly;
specify what is taken as given by each agent; and which markets clear.
(c): Solve for the equilibrium. Note that all variables will be functions of g and z; which are the
exogenous variables for which we didn t take speci
(d): Analyze how a decrease in total factor productivity a⁄ects the equilibrium levels of C;l and
w: To answer this question use the expressions for consumption, leisure and the wage that you have
derived in point (c): If necessary, take derivatives with respect to z to infer your answer. Explain
the e⁄ects of a decrease in z on C and l using income and substitution e⁄ects.
(e): Analyze how an increase in government spending a⁄ects the equilibrium. Explain.
(a): An exogenous variable is one determined outside of the model.
An endogenous variable is determined using the model s equilibrium conditions and the values
of the exogenous variables.
(b): Given exogenous variables fg;zg; a competitive equilibrium (C.E.) is a set of endoge-
▯ S d
nous quantities C;l;N ;N ;▯;Y;T;G , and an endogenous price w that satisfy the following:
▯ Given the wage w; the dividend income ▯ and taxes T; the consumer chooses C;l and N to
maximize utility subject to constraints.
maxflnC + 0:02 ▯ lg
C = wN + ▯ ▯ T
N + l = h = 100
▯ Given the wage w; and exogenous variable z, the
rm chooses N to maximize pro
n ▯ d0:7 do
max 3 ▯ z ▯ N ▯ w ▯ N
▯ The government budget is balanced: T = G = gY
▯ Markets clear:
Labor market: N = N
Goods market: Y = C + G or C + gY = Y or C = (1 ▯ g)Y
(c): Solve for the equilibrium. This means
nding expressions for C;l;N ;N ;▯;Y;T;w which
depend on exogenous variables only!
MRS l;C w ) 0:02 ▯ C = w (1)
w = 3 ▯ 0:7 ▯ z N
w = 2:1 ▯ z N (2)
Equate equations (1) and (2):
0:02 ▯ C = 2:1 ▯ zN
2 Get rid of C in equati